Cloth particles: Seeking workflow
With the cloth simulation, you want to select your dollar bill object and go to physics tab; it's in the Properties window under the Outliner by default, same window you go to for your particle systems. The icon is a bouncing ball. If you can't see it, over over the panel and scroll until it's in sight, or enlarge the window. Hit the cloth button. You're going to want to play around with the stiffness setting in particular. You don't want it to act like a dollar-printed hankie. Once you have setting to work with, you'll need to let it run it's simulation. You'll probably do this several times until you get something you like. After you're finally satisfied, bake all your dynamics.

Go back to your particle system; make sure you have gravity on. Mess with the weight values; your money could just be too light. Finally, apply the bill's orientation so that the origin is in the middle of the bill, and it lays flat on the floor by default. That part should help with the standing problem, at least.